### Appendix 13.1: Derivation of the Relative Gain for an n-Input–n-Output System

The relative gain between input j and output i is defined as

Output i is the following steady-state function of all of the inputs:

The relationship between input j and output i with all other inputs constant is

For any desired output vector, y, we can find the corresponding input vector, u, from

Defining Ĝ(0)=G^{-1}(0)=K^{-1}

The relationship between all outputs and input ...